Two calculus students attempt to “break” math.

**Work in groups of 3–4, writing your answers on a separate sheet of
paper.**

Check out this dialogue between two calculus students (based on a true story):

- Devyn
- Riley, I have something very important to say.
- Riley
- Yeah? Hit me with it.
- Devyn
- I think I just broke math.
- Riley
- I’ve suspected for ages that all this calculus stuff was razzmatazz. Lay it on me.
- Devyn
- Consider the vector field:
- Riley
- Got it. It looks like a whirlpool.
- Devyn
- I know! Now compute its curl.
- Riley
- OK–I get zero curl.
- Devyn
- I know! Now consider Green’s Theorem.
- Riley
- You mean: What’s ? What’s ?
- Devyn
- Let be the unit disk centered at the origin.
- Riley
- OK, so is the unit circle centered at the origin.
- Devyn
- Right. Now here’s the deal…The left-hand side of the equation is zero because the curl of our vector field is zero.
- Riley
- Oh. And the right-hand side of the equation cannot be zero because our vector field looks like a whirlpool.
- Devyn
- And now we have zero equals something not zero, and kablamy, math is in ruins.

Let’s investigate further. To really understand this, we’re going to have to check each of their claims.

**interior**. A square of some sort would probably be easiest.

At this point, you might suspect that something strange is going on…

### The take-away

Here we presented you with a field where the (scalar) curl was zero everywhere except at the origin. At the origin the (scalar) curl was undefined; hence, Green’s Theorem does not apply.

What is remarkable is that in this case, where the (scalar) curl is zero except for a point, any path around the point where the field is undefined will yield the same value for: